How to Decide on Airport Location from a Welfare
Perspective
A Theoretical approach using Spatial Economic Theory
Author: Erik Johansson
Supervisor: Jonas Westin
Student
Spring 2014
Master Thesis I, 15 ECTS
Master's Program in Economics
Acknowledgements
Spatial economics is a eld that is much larger than I could ever
imagine when I decided to write this paper. It is a part of the
economic discipline that I had only briey encountered, but there
are ever more texts about it the deeper you dig.
Therefore I
would like to thank some of those who helped me and pushed me
forward during this work. My supervisor, Jonas Westin, deserves
many thanks for his advices, positiv thinking and endless ow of
ideas. I should also mention my classmates, writing on their own
theses, for sharing the burden of anxiety. Last, but not least, the
opponents on the seminar deserves appreciation for their valuable
comments.
Abstract
This paper uses a location theory inspired by Hotelling and Launhardt to demonstrate the optimal location and size of airports
from the viewpoint of the social planner.
The method used is
purely analytical where a model is set up and then analysed.
The ndings show that the optimal size for one airport in the
single airport case is larger than the optimal size for one airport
in the dubble airport case. In addition, for low transport costs
(t), the cost of travel from the point of origin to the airport, it
is favourable to have one airport and for high
to have two airports. The boundary point of
t
t
it is favourable
where the social
surplus functions are equal is derived explicitly.
Keywords : Airports, Spatial Competition, Location Theory
Contents
1
Introduction
1
1.1
The Airport Market . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2
Theoretical Framework
5
3
The Model
8
4
Model Analysis
11
4.1
Deriving social surpluses . . . . . . . . . . . . . . . . . . . . . . . . .
11
4.2
Optimal location and service level . . . . . . . . . . . . . . . . . . . .
14
4.3
Transport cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4.3.1
Comparative statics . . . . . . . . . . . . . . . . . . . . . . . .
17
4.3.2
Geometric analysis
18
. . . . . . . . . . . . . . . . . . . . . . . .
5
Concluding Remarks
20
6
References
23
A Mathematical Appendix
26
1
Introduction
The question of airports is these days a hot topic with the debate about the existence
of Bromma Airport in Stockholm, a new runway at one of the London airports and
not to forget the failure of the projected Berlin Brandenburg Airport in Germany,
planned to open in 2011, but still struggling.
The reason why these issues are so
heavily debated is because airports are an important part of a country's infrastructure
to make distant areas accessible and airports are also subject to regional and local
prestige. Cities with an airport, or even better a major airport, are visited by people
and ideas from other places and the inhabitants can easily reach distant regions. In
northern Sweden and other remote and scarcely populated areas, there is an issue
of deciding which airport that should be in scope for government subsidized ights.
These ights are of great importance to the people in those communities enabling
them to reach the capital in a reasonable time. In contrast to those remote airports
there is the question of where the next runway should be built in the Greater London
area. Although these issues are dierent they are still related: which airport do we
want people to travel from?
In the Swedish case the demand is too low for two
(relatively) close airports and in the London case the demand is too high for the
available facilities.
This paper will address these issues from a theoretical point of view using a
classical location theory which is related to spatial economic theory.
This theory
started with Johann Heinrich von Thünen and his work The Isolated State (1826)
where he studied the land use of dierent activities surrounding a town, e.g. milk
production. Without going further there is a need to clarify some central concepts
in spatial economic theory and Puu (2003) put it like this: There are two basically
dierent ways of dealing with the spatially extended economy.
1
We can ask the
question:
`Where should a certain activity be located?'; or we can ask:
`Which
activity should be chosen at a certain location?'. Location theory deals with the rst
question, land use theory with the second. Both, of course, address the principles
underlying the spatial layout of an economy.
The purpose of this paper is not to come to a solution whether to extend Heathrow
or Gatwick but to propose that location theory can be applied to competing airports.
The point will be made with a simple theoretical model similar to the one suggested
by Harold Hotelling in Stability in Competition (1929).
The research question
used as a basis for this analysis is Will one or two airports be better from a welfare
perspective?
1.1
The Airport Market
The global airport market was until the mid-80s characterized as a publicly controlled
business but starting with the privatization of the British Airport Authority (BAA)
in 1986, the market is nowadays to a large extent privately or partly privately owned.
Odoni (2009) points out six factors that were the driving forces in this development,
among them the argument that the airport business had become a mature industry
in the sense that it could be economically self-sucient. Further, non-aeronautical
businesses developed to be a signicant source of revenue to airports, and also, as
the aviation industry grew, there was a demand for another management of the
airports that could operate with more exibility and economic eciency. Moreover,
he states that the privatization often is realized through a build, operate and transfer
(BOT) contract. The authority signs a contract over several (perhaps many) years
with a company or a consortium of companies and acquire some shares.
This is
also known as a public-private partnership (PPP) where the government and the
2
legislative assembly arrange for a private business to do the job (for more on airport
property see, for example, Guidebook for Developing and Leasing Airport Property
(2011)).
Furthermore, a large body of literature has been devoted to congestion
at airports, such as Pels & Verhoef (2004), Brueckner (2005) and Silva & Verhoef
(2013), which is an issue mostly at major airports serving large cities or operating
as hubs. However, this paper will not contribute in this aspect.
IATA (International Air Transport Association) stated that more than 60 percent of the European population live within two hours from at least two airports
(Wiltshere, 2013).
In Sweden, 98 % of the population have access to at least one
airport within two hours of connecting transport mode (Transport Analysis, 2013).
The author of the IATA report referred to a bulletin that examined the competition
between a few English airports using isochrones around the airports to assess the
regions of competition and could recognize several signicant areas of overlapping
isochrones (Frontier Economics, 2008).
1
This indicates that the airport market is
to some extent characterised as spatially monopolistic, meaning that each airport
enjoys some market power in their region (Pels and Verhoef, 2004). This notion is
supported by the bulletin which states that 1 % increase in distance to a certain airport implies a 4 % decrease in the likelihood of traveling from that particular airport
on average, and a 1 % increase in distance must be oset by a percentage decrease
in the relative price. Furthermore, Zhang and Xie (2005) studied the inuences of
airport choice in small cities and concluded that ticket price, distance to airport and
the time of arrival and departure were the most important factors. In Sweden the
airports are to a large extent operated by the public or by the government owned
company Swedavia. The only private agent is Aviation Capacity Resources (ACR)
1 Isochrones are commonly used to describe travel time to a certain place and looks like a jagged
circle on a map.
3
who runs nine regional airports. Many airports, however, perform badly economically, especially airports operating as spokes (Transport Analysis, 2014).
Bråthen
(2011) and Transport Analysis (2013) argue that airports enjoy economies of scale
and the market may then be characterized as a natural monopoly where a price
equal to marginal costs would lead to nancial problems because of the low marginal
costs and high xed costs (Hansen and Nerhagen, 2008). Another concern is the case
of externalities which may not be taken into account by a prot maximizing rm.
This, together with the fact that the airport can be characterized by monopolistic
competition are two arguments for regulation of airports in order to internalize the
social welfare aspect and externalities.
1.2
Thesis structure
The paper is divided into ve sections. In section two the theoretical underpinnings
are presented and the evolution of spatial economics is briey explained. In the third
section the model is built together with the underlying assumptions and in section
four the model is analysed, starting with deriving the social surpluses, continuing
with the equilibrium locations and service levels and eventually ending with an analysis and interpretation of the transport costs. Lastly there is a section for discussion
and concluding remarks.
Throughout the analysis two cases are studied: the case of two airports and the
case of one airport. Note that the most technical parts are included in the appendix,
leaving the core in the main text.
4
2
Theoretical Framework
The theory of spatial economics is said to have started with Thünen in the early 19th
century (Fujita, 2010). Thünen (1826) observed how dierent activities were located
around towns with respect to land rent and transportation costs. Samuelson (1983)
marked the 200th birthday of Thünen and spared no words when he described him:
Thünen belongs to the Pantheon with Léon Walras, John Stuart Mill and Adam
Smith. As Schumpeter would say, it is the inner ring of Valhalla they occupy. After
Thünen (many years after) came more men with beard or moustaches or neither
of it and developed their own theories but there was, until the middle of the 20th
century, not much coherence between the English speaking academic world and the
German counterpart, mainly for linguistic reasons. The German speaking literature
consisted of Thünen (1826), Launhardt (1885), Weber (1909), Palander (1935) and
Lösch (1940) etc.
and the English of Marshall (1890) and Hotelling (1929) etc.
The eld is so vast that one could write a paper only about the development of
it, and it was done by Fujita (2010) who very thoroughly recognize the important
works from von Thünen to present day. Isard (1949) was perhaps one of the rst to
summarize what Launhardt, Weber and others had done and critisized economists
for not considering space in economic theory. This was also highlighted by Krugman
(1991a) where he stated that economic geography plays at best a marginal role in
economic theory.
The base theory of this paper will be similar to Hotelling's model (1929).
theory describes the location of two competing rms along a line, (e.g.
His
ice cream
sellers along a beach). The consumers will buy from the seller that entail the lowest total cost, that is, the actual price of the good plus the cost of transportation
(Hotelling, 1929).
He argues that a rm cannot increase the price without losing
5
business to the rival, i.e. the marginal consumer who is indierent to the sellers will
shift. Hotelling's conclusion is that the sellers will locate right next to each other and
the analogous reasoning goes for rms producing similar products, political parties
trying to maximise their vote share and so on. This concept is known as the Principle of Minimum Dierentiation. However, Ferreira and Thisse (1995) argues that
Wilhelm Launhardt deserves credit for his work Mathematische Begründung der
Volkswirtschaftslehre in 1885 which is a special case of Hotelling's model written 44
years earlier (Fujita, 2010).
2
Returning to Hotelling, he makes one brave assumption about inelastic demand
leading to a case where it is costless for the seller to move away from the hinterland
and towards the centre. One criticism to this paper comes from d'Aspremont et. al.
(1979) where an argument for the opposite case is made, namely that there is no
tendency to minimum dierentiation; it proved that with quadratic transportation
costs the rms will locate at the endpoints of the line (Maximum Dierentiation).
Their argument is that there cannot be an equilibrium price if the sellers are located
too close to each other.
Sun and Lai (2013) shed new light on the debate when they applied Hotelling's
case to a situation with decreasing returns to scale and argue that the principle of
minimum dierentiation can be conrmed, that is, rm agglomeration at the centre
of the market is the unique location equilibrium.
Smithies (1941) makes a verbal
argument on Hotelling's case. He states that producers can steal consumers either
by cutting the price or nding a new location.
The basic assumption is a linear,
one dimensional market where there are two producers of a certain product. They
can locate along this line and supplies the share of consumers who minimizes their
expenditure by buying their products. That is, they will supply all the consumers
2 The English version of the book is Principles of Mathematical Economics (1993)
6
in their hinterland (away from the centre) and the consumers on their side of the
boundary point, which in the case of symmetry will be right in the middle of the
line. If one of the producers change either price or location the boundary point will
shift in the direction implied by the preceding change. Smithies (1941) summarizes
Hotelling's case by mentioning that the supplier can, and will, move to the centre
of the market and pass on the increased freight and travel charges to the consumers
without aecting prots. To get the full picture of the theory one should look at it
as in gure 1. Illustrated are two completely arbitrary cases which also happens to
be symmetrical. The solid line applies for two airports, or whatever one is studying,
located at equal distance from the centre and the dashed line applies for one airport.
The y-axis shows the costs, the closer to (further from) an airport, the lower (higher)
the costs. The x-axis illustrates the market or geographical plane along which people
live. This is the gross price that the consumers face.
p
0
1/2
1
3/2
Figure 1.A Hotelling model with two firms (solid) and one
firm (dashed)
7
2
x
3
The Model
In this section the model is presented and in the next it is to be examined analytically.
Assume two municipalities (they will hereafter be called towns) located along a line
∈ [0, 2] where people live uniformly and continuously.
is located in the middle meaning that
denotes the towns.
A ∈ [0, 1]
The border between the towns
and
B ∈ [1, 2],
where A and B
The towns are assumed to be symmetric in all other senses.
In addition to the towns there is a federal decision level that encompass both A
and B. Furthermore, the airport(s) is (are) constrained to supply the entire demand
and the towns as well as the federal state are assumed to be welfare maximizing.
The towns only consider the welfare for people living within its borders meaning
that they maximize welfare for their inhabitants, whereas the federal government
consider both towns' inhabitants when maximizing welfare.
Transportation costs
(to connect the travellers to the airports) are proportional to distance. Hence, the
cost of travelling between any two points along the line is the same. Throughut the
analysis subscript
j
will denote airports (if the airports are not specied as either A,
B or F). Superscripts in conjunction with
SS
or
TS
for example denotes the number
of airports.
3
Now, suppose for simplicity that the demand for airports is constant for all prices.
q = q (x)
The total demand for town A and B is equal to the integral of their territory
respectively and the demanded quantity on the federal level equals the sum of A and
B.
3 This assumption is supported by Basso (2008).
8
ˆ
x̂
q (x) dx = q · x̂
(1a)
q (x) dx = q · (2 − x̂)
(1b)
QA =
0
ˆ
2
QB =
x̂
QF = QA + QB = 2 · q
(1c)
The rst two equations are valid for two airports where
x̂
is the boundary point,
that is, the point where the marginal traveller is indierent between the two airports.
The third equation is valid for the case with one single airport. Thus, the demand for
each airport must be the integral from the edge of the market area to the boundary
point. Moreover, inspired by Ferreira and Thisse (1995), the travellers use the airport
that impose the lowest expenditure
p + t · |x − xj |
p
(2)
is the airfare, assumed equal for both airports (since the price depends on the
4
airline pricing which is not witin the scope of this paper) ,
cost,
x
is the place of origin of each individual and
xj
t
is the transportation
is the location of airport
j.
When deciding which airport to use the travellers also takes into account variables
such as service. Then the traveller's surplus (tsi ) can be written as
tsi (x) = q · (z + sj − p − t · |x − xj |)
The term
z
(3)
is a general utility of travelling (assumed equal for all travellers) and
4 For studies concering airline behaviour, see e.g. Escobari and Lee (2012).
9
sj
is the service level and size of the airport.
We will now dene the boundary point described above. The fact that the trav-
ellers are indierent between the airports (at this point) must imply that
tsB (x̂).
Using equation (3) above and solving for
Proposition 1. The boundary point,
x̂,
x̂
tsA (x̂) =
yields the following proposition
is where the marginal traveller is indif-
ferent between the airports.
1 1
· (sA − sB ) + (xA + xB )
x̂ = ·
2
t
(4)
Proof.
q · (z + sA − p − t · (x̂ − xA )) = q · (z + sB − p − t · (xB − x̂))
2 · t · x̂ = sA − sB + t · xA + t · xB
1 1
x̂ = ·
· (sA − sB ) + (xA + xB )
2
t
Substituting the equation for
QA = q ·
QB = −q ·
x̂ into equation (1a) and (1b) respectively produces
1 1
·
· (sA − sB ) + (xA + xB )
2
t
1 1
·
· (sA − sB ) + (xA + xB ) − 2
2
t
10
In addition, the costs,
costs
V,
C,
that is dependent on the usage of the airport
of the airport,
saj ,
returns to scale,
where the parameter
a
Q,
F,
variable
as well as the size/service
is a scaling factor.
a=1
implies constant
a > 1 implies increasing returns to scale and a < 1 means decreasing
returns to scale. In this paper
of scale.
of the airport are divided into xed costs
a is assumed to be equal to 2, which implies economies
5
Cj = (F + V · Qj ) · s2j
Furthermore, the airports are assumed to receives a fraction
(5)
θ
of the airlines'
revenues which, like the price, is assumed to be exogenously given and equal for all
airports. This can be realized by imposing a per-ight toll or a charge per passenger
(Silva and Verhoef, 2013). Let
R
stand for the revenues as follows
Rj = θ · Qj · p
4
(6)
Model Analysis
4.1
Deriving social surpluses
The purpose of this subsection is to derive the social surpluses that are subject to
further analysis in the latter parts of section 4. Beginning with the two airport case,
5 This assumption is supported by Bråthen (2011) and Transport Analysis (2013). Also, after
some trial and error, where
a = a
was used, the resulting expressions were unclear and rather
unrealistic.
11
integrate equation (3) above, that is, the traveller's surplus (TS), for town A and
B respectively. Suppose that the boundary point is located o centre so that some
people in one of the towns use the other town's airport. Here
x̂ > 1 is used, meaning
that some indivduals in town B will use the airport in town A.
ˆ
T SA2
ˆ
xA
q (z + sA − p − t · (x − xA ))
xA
0
=
1
q (z + sA − p − t · (xA − x)) +
=
1
1
(q · xA (2 · p − 2 · sA − 2 · z + t · xA )) − · (q · (xA − 1) (2 · sA − 2 · p − t + 2 · z))
2
2
(7a)
ˆ
T SB2
=
1
ˆ
x̂
xB
q (z + sB − p − t · (xB − x))
q (z + sA − p − t · (x − xA )) +
x̂
ˆ 2
q (z + sB − p − t · (x − xB ))
+
xB
=
1
· q · (sA − sB − 2 · t + t · xA + t · xB ) (3 · sA − 4 · p + sB − 2 · t + 4 · z + 3 · t · xA − t · xB )
8·t
(7b)
1
− · q · (xB − 2) (2 · sB − 2 · p − 2 · t + 2 · z + t · xB )
2
−
1
· q · (sA − sB + t · xA − t · xB ) (sA − 4 · p + 3 · sB + 4 · z + t · xA − t · xB )
8·t
Switching focus to the case where one airport supplies the entire demand and
integrate equation (3).
Assume that the airport is located in town B. Notice the
12
dierence between superscript 1 and 2, which denotes one airport and two airports
respectively.
ˆ
T SA1
1
q · (z + sF − t · (xF − x) − p) · dx
=
0
T SA1 =
ˆ
T SB1
1
· q · t − q · (p − sF − z + t · xF )
2
ˆ
xF
2
q ·(z + sF − t · (xF − x) − p) · dx+
=
1
(8a)
q ·(z + sF − t · (x − xF ) − p) · dx
xF
T SB1 =
1
· q · (xF − 1) (2 · sF − 2 · p + t + s · z − t · xF )
2
(8b)
1
− · q · (xF − 2) (2 · sF − 2 · p − 2 · t + 2 · z + t · xF )
2
Using equation (5) and (6) from the previous section the net costs can be dened
as
N C j = Cj − R j
Now, to derive the social surpluses the net costs are subtracted from the traveller
surplus for each town individually. For the case of one airport in each town, they
pay the full costs for operating one airport each and for the case of one joint airport
the towns pay one half of the costs each. Further, when the local social surpluses are
derived according to equation (9) they will be summed according to equation (10)
below, generating the federal social surplus
13
SSF .
There will be one federal social
surplus function for each case, denoted
SSF1
and
SSF2 .
SS = T S − N C
SSF = SSA + SSB
(9)
(10)
As mentioned in the in the introduction the social surplus functions are presented
in the appendix to make the analysis more readable.
4.2
Optimal location and service level
Dierentiating the federal social surplus functions with respect to the locations and
service levels produces the following propositions
Proposition 2.
Optimal locations of two airports
xA =
1
2
xB =
3
2
Optimal location of one airport
xF = 1
14
Proof. Dierentiate the social surplus function for two airports (eq. 11 in Appendix) w.r.t.
xA
Appendix) w.r.t.
and
xF
xB
and the social surplus function for one airport (eq. 13 in
and evaluate at 0.
∂SSF2
1
= · q · sA − sB − 3 · t · xA + t · xB − V · s2A + V · s2B = 0
∂xA
2
∂SSF2
1
= · q · sA − sB + 4 · t + t · xA − 3 · t · xB − V · s2A + V · s2B = 0
∂xB
2
∂SSF1
= −2 · q · t · (xF − 1) = 0
∂xF
xA ,
Using algebra one can solve the optimal
xA
substitute into the second and solve for
xB .
into the function for
xA
and solve for
xA .
and
xB
by solving the rst FOC for
Take the optimal
Then, assuming that
xB
and substitute
s = sA = sB
yields
6
the proposed results. The second derivatives shows that this is a maximum.
Proposition 3.
The optimal service level for one single airport is larger than
the optimal service level for one airport in the dual airport case.
s = sA = sB =
sF =
q
(2 · F + 2 · V · q)
q
(F + 2 · V · q)
6 The second derivatives and decision rules are presented in the Appendix.
15
sA = sB < sF
Proof.
Taking the derivative of
(assuming that
s = sA = sB
and
SSF2
(eq.
12 in Appendix) with respect to
xA = 2 − xB )
s
and evaluating it at its optimimum
produces
∂SSF2
=2·q−4·F ·s−4·V ·q·s=0
∂s
Dierentiating
SSF1
(eq.
13 in Appendix) w.r.t.
to
sF
and evaluate at optimum
yields
∂SSF1
= 2 · q − sF · (2 · F + 4 · V · q) = 0
∂sF
Both optima are also maxima when looking at the second derivatives, which are
7
negative.
The results from proposition 2 and 3 can be summarized by gure 2a and 2b
illustrating the higher service level as a downshift of the curve.
travel cost
travel cost
0
1/2
1
3/2
2
x
0
Figure 2a.
Figure 2b.
7 The second derivatives are presented in the Appendix.
16
1
2
x
4.3
Transport cost
4.3.1
Comparative statics
The objective now is to analyse the dierence between the federal social surpluses,
SSF1
and
SSF2 ,
and investigate this dierence when dierentiating with respect to
the transport cost.
following way:
SSF1
A negative sign of the derivative should be interpreted in the
an increase in transport costs by a marginal amount implies that
becomes relatively larger than
be interpreted as an increase in
relatively smaller than
SSF2 .
t
SSF2 .
A positive sign of the derivative should
by a marginal amount implies that
SSF1
becomes
Thus, in the rst case (a negative derivative) it becomes
more benecial to have two airports as the transport costs increase and in the latter
case (a positive derivative) it becomes more benecial to have one airport as the
transport costs increase.
Let
DW = SSF1 − SSF2
and subsitute
xF , xA , xB , s F , s A
and
sB
values (from proposition 2 and 3) and dierentiate with respect to
Proposition 4. For low values of
high levels of
t
t
for their optimal
t.
it is favourable to have one airport and for
it is favourable to have two airports.
Proof.
∂DW
q · (F + 3 · F · V · q + 2 · V 2 · q 2 )
=−
<0
∂t
2 · (F + V · q) (F + 2 · V · q)
∂ 2 DW
=0
∂t2
17
The derivative is negative and a look at the second derivative shows that
negative for all values of
t.
DW
is
The intuition is obvious, if transportation costs are high,
people want short distances to the airport. Illustratively, this would be equivalent to
atter curves in the gures above for low values of
of
t.
t and steeper curves for high values
Antother way to illustrate this is to form a diagram with
axis and the derivative
∂DW/∂t
t
on the horizontal
on the vertical axis and draw the curve downward
sloping for all values of t. From this reasoning about the rst and second derivatives
there must be a value of
t = t̂ where the dierence is zero, that is, where it is equally
favourable to have one airport and two airports. Solving
t̂ =
4.3.2
DW
for
t
gives
q·F
2 · q2 · V 2 + 3 · q · F · V + F 2
Geometric analysis
Suppose that one can divide the travellers' surplus into two parts; one part related
to the connection to the airport
directly related to the airport,
T STj ,
and the other part related to the experience
T SSj .
T Sj = T STj + T SSj
Then the social surplus ( from equation 9) can be written
SSj = T STj + T SSj − N C
By adding the local social surpluses and then multiplying by
(−1)
illustrated by a diagram very similar to the the previous ones. Here is the
this can be
N C −T SS
assumed equal for both cases, meaning that the focus is on the connection to the
airports. By evaluating the the triangles one can compare the total travel costs that
18
the travellers face with the dierent number of airports.
t
a
b
d
c
TST1
TST2
NC-TSS
0
1/2
1
3/2
2
x
Figure 3.
Two airports
1
2
· 12 · t
t
·4·q = ·q
2
2
One airport
t·1
·2·q =t·q
2
Thus, the total travel cost in the case of two airports is only a half of the travel
cost with one airport.
Another way to interpret this gure is to look at it as how people would vote in a
hypothetical referendum, given their location along this line. Under the assumption
8
that they will cast their vote on the alternative that imply the lowest costs it becomes
clear that people living at 0-0.75 and 1.25-2 would prefer two airports and those living
at 0.75-1.25 would prefer one airport, given this rate of travel cost. Why? As one
can see, the dashed line, symbolising the one airport case, exceeds the solid line at
values of
x
less than 0.75 and more than 1.25 meaning that the transport costs of
8 Hence, pure self-interest voting.
19
having one airport exceeds the costs with two airports. The dierence in aggregated
costs can be illustrated by the areas
a+d
and
b + c.
pictured in gure 2a and 2b it is evident that the areas
and
c
However, using the results
a and d will be smaller and b
will be larger since the service level of a joint airport exceeds the service level
of an airport in the dual airport case. As gure 4 illustrates, the boundary points,
where people are indierent between one and two airports, would shift towards the
ends of the line and thus imply that relatively more people would be in favour of the
one airport case. One can see, by analysing the vertical distance between the solid
line and the dashed line, that a few individuals (those living close to the border)
will gain much with one airport while most individuals will experience slightly more
disutility in this case.
{
a
TST1
{NC-TSS
b
d
c
TST2
{x
NC-TSS
1
0
{
1/2
1
3/2
2
2
Figure 4.
5
Concluding Remarks
A goal of this paper is to contribute to the literature that use location theory and to
summarize the straggly literature that is spatial economics and it is arguably a eld
of economics that has been ignored (perhaps unconsciously) from time to time.
The purpose was to answer the question if one or two airports would be better
20
from a welfare perspective and todo this a simplied model was proposed. Proposition 2 in section 4.2 states that one single airport will have a higher optimal service
level than an airport in the dual airport case. Although two airports would have a
higher aggregated service level this would in reality imply that they provide the same
routes at the same time, for example ights to Stockholm on weekday mornings in
a Swedish context. The policy implication is that, as Bråthen (2011) mentions, the
federal government could do a better job by focusing the resources on one airport
and improve the connection possibilities to and from the airport.
As a result the
region could attract a larger variety of carriers and also gain better accessability to
other regions via direct ights.
Further conclusions depends on how large the transport costs are. As depicted
in proposition 4 in section 4.3.2, high transportation costs points in the direction of
two airports. Analogously, low transportation costs would imply that one airport is
better. The key is the euation for
t̂
on page 19.
This study does not acount for eects like congestion and externalities, which in
the real world are of great importance. It is reasonable to suppose that congestion
issues would push in the direction of the two aiport case because the aggregated
service level could then be higher. This issue is indeed relevant for large cities like
London.
On the other hand, negative environmental externalities, e.g.
and noise, would arguably point in the direction of one airport.
pollution
Moreover, this
analysis does not take into account the topographical features of the real world. For
example in countries like Norway it is simply not possible to establish an airport at
all locations due to mountains and fjords, a so called geographical constraint. Thus,
the theoretically optimal location may not coincide with a feasible location.
Further conclusions are dicult to make from this paper and it should be clear
that much more work can be done on this topic it but the model does illustrate that
21
location theory can be used to analyse airport location. As a closing thought it will
be interesting to follow up future airport investments and closures and to see when
spatial economics will be a comprehensive theory in textbooks used by students.
22
6
References
Basso, L. (2007).
Airport deregulation: Eects on pricing and capacity.
Interna-
tional Journal of Industrial Organization 26: 1015-1031.
Brueckner, J. K. (2005). Internalization of airport congestion: A network analysis.
International Journal of Industrial Organization 23 (7/8): 599-614.
Bråthen, S. (2011). Air Transport Services in Remote Regions. International Transport Forum. Discussion Paper No. 2011-13.
Chiang, A. C. and Wainwright, K. 2005.
Fundamental Methods of Mathematical
Economics. 4th ed. New York: McGraw-Hill.
d'Aspremont, C., Gabszewicz, J. J. and Thisse, J.-F. (1979). On Hotelling's Stability
in Competition. Econometrica 47 (5): 1145-1150.
Escobari, D. and Lee, S.-Y. (2012). Demand shifting across ights and airports in a
spatial competition model. Letters in Spatial and Resource Sciences 5: 175-183.
Ferreira, R., Dos Santos and Thisse, J.-F. (1995). Horizontal and vertical dierentiation: The Launhardt model. Internationl Journal of Industrial Organization 14:
485-506.
Frontier Economics (2008), Keeping the lid on. Analysing competition between UK
airports.
Fujita, M. (2010). Evolution of Spatial Economics: From Thünen to the New Economic Geography. The Japanese Economic Review 61 (1).
Hansen, F. and Nerhagen, L. 2008. Swedish airports and marginal cost pricing. VTI
rapport 633. The Swedish National Road and Transport Research Institute.
23
Hotelling, H. (1929).
Stability in Competition.
The Economic Journal 39 (153):
41-57.
Isard, W. (1949). The General Theory of Location and Space-Economy. Quarterly
Journal of Economics 63: 476-506.
Krugman, P. (1991a).
Increaing Returns and Economic Geography.
Journal of
Political Economy 99: 483-499.
Launhardt, W. (1885). Mathematische Begründung der Volkswirtschaftslehre. Leipzig:
B.G. Teubner.
Reprinted in English translation (by H. Schmidt) as Mathematical
Principles of Economics, Aldershot, Edward Elgar, 1993.
Lin, M. H. and Kawasaki, A. (2010), Where to enter in hub-spoke airline networks.
Papers in Regional Science 91 (2): 419-436.
Lösch, A. 1940.
Die Räumliche Ordnung der Wirtschaft.
Jena:
Gistav Fischer.
Translated by Woglom W. H. 1954. The Economics of Location, New Haven, CN:
Yale University Press.
Marshall, A. 1890. Princplies of Economics. London: Macmillan.
Odoni, A., 2009.
The international institutional and regulatory environment.
In:
Belobaba, Peter and Odoni, Amedeo and Barnhart, Cynthia (Eds.), The Global
Airline Industry (John Wiley & Sons).
Palander, T. (1935). Beiträge zur Standortstheorie. (Doctoral dissertation). Uppsala: Almqvist & Wiksell.
Pels, E. Verhoef, E. T. (2004). The Economics of Airport Congestion Pricing. Jour-
nal of Urban Economics 55 (2): 257-277.
Puu, T., 2003. Mathematical Location and Land Use Theory: An Introduction. 2nd
ed. Berlin: Springer-Verlag.
24
Samuelson, P. A. (1983). Thünen at Two Hundred. Journal of Economic Literature
21: 1468-1488.
Silva, H. E. and Verhoef, E. T. (2013). Optimal pricing of ights and passenger at
congested airports and the eciency of atomistic charges.
Journal of Public Eco-
nomics 106: 1-13.
Smithies, A. (1941). Optimum Location in Spatial Competition. Journal of Political
Economy 49 (3): 423-439.
Sun, C.-H. and Lai, F.-C. (2013). Hotelling was right with decreasing retuns to scale
and a coalition-proof renement. Annals of Regional Science 50: 953-971.
Thünen, J. H. von. 1826. Der Isolierte Staat in Beziehung auf Landwirtschaft und
Nationalökonomie. Hamburg: Perthes. Translated by Wartenberg C. M. 1966. Von
Thünen's Isolated State, Oxford: Pergamon Press.
Transport Analysis (2014): Civil aviation tables 2013, Statistik 2014:3, Stockholm.
Transport Analysis (2013): Flygplatsers funktion och sårbarhet, Rapport 2013:12,
Stockholm
Transport Analysis (2014): Konsekvenser av regleringsförändringar på ygtraktjänstområdet, Rapport 2014:1, Stockholm
Weber, A. 1909. Über den Standort der Industrien, Tübingen: J. C. B. Mohr. Translated by Friedrich, C. J. 1929. The Theory of the Location of Industries, Chicago:
Chicago University Press.
Wilthsere, James (2013). Airport Competition. International Air Transport Association (IATA). IATA Economics Brieng: 11.
Zhang, Y. and Xie, Y. (2005). Small community airport choice behavior analysis: A
case study of GTR. Journal of Air Transport Management 11: 442-447.
25
A
Mathematical Appendix
Social Surpluses
Social surpluses for town A and B given two airports.
SSA2 =
1
· p · q · θ · (sA − sB + t · xA + t · xB )
2·t
1
− · q · xA · (2 · p − 2 · sA − 2 · z + t · xA )
2
1
− · q · (xA − 1) (2 · sA − 2 · p − t + 2 · z + t · xA )
2
1
−
· s2A · (F + V · q · (sA − sB + t · xA + t · xB ))
2·t
SSB2 =
1
· q · (sA − sB − 2 · t + t · xA + t · xB )
8·t
· (3 · sA − 4 · p + sB − 2 · t + 4 · z + 3 · t · xA − t · xB )
1
− · q · (xB − 2) (2 · sB − 2 · p − 2 · t + 2 · z + t · xB )
2
1
2
−sB ·
· (sA − sB + t · xA + t · xB ) − 2
2·t
1
−p · q · θ ·
· (sA − sB + t · xA + t · xB ) − 2
2·t
−
1
· q · (sA − sB + t · xA − t · xB ) (sA − 4 · p + 3 · sB + 4 · z + t · xA − t · xB )
8·t
26
Social surplus for the federal government given two airports.
SSF2 =
(11)
1
· q · (sA − sB − 2 · t + t · xA + t · xB ) (3 · sA − 4 · p + sB − 2 · t + 4 · z + 3 · t · xA − t · xB )
8·t
1
2
−sB ·
· (sA − sB + t · xA + t · xB ) − 2
2·t
1
− · q · (xB − 2) (2 · sB − 2 · p − 2 · t + 2 · z + t · xB )
2
1
−p · q · θ ·
· (sA − sB + t · xA + t · xB ) − 2
2·t
1
· q · (sA − sB + t · xA − t · xB ) (sA − 4 · p + 3 · sB + 4 · z + t · xA − t · xB )
−
8·t
1
− · q · xA · (2 · p − 2 · sA − 2 · z + t · xA )
2
1
− · q · (xA − 1) (2 · sA − 2 · p − t + 2 · z + t · xA )
2
1
· s2 · (F + V · q · (sA − sB + t · xA + t · xB ))
−
2·t A
1
+
· p · q · θ · (sA − sB + t · xA + t · xB )
2·t
27
Assuming that
SSF2
sA = sB = s
and
xA = 2 − x B
generates
1
1
2
=
· p · q · θ · (t · xA − t · (xA − 2)) − s ·
· (F − V · q · (t · xA − t · (xA − 2))) − 2
2·t
2·t
(12)
1
1
− · q · xA · (2 · p − 2 · s − 2 · z + t · xA ) − · q · (xA − 1) (2 · s − t + 2 · z + t · xA )
2
2
1
− · q · xA · (2 · p − 2 · s + 2 · t − 2 · z + t · (xA − 2))
2
1
−
· q · (2 · t − t · xA + t · (xA − 2)) (4 · s − 4 · p − 2 · t + 4 · z + 3 · txA + t · (xA − 2))
8·t
1
· (t · xA − t · (xA − 2)) − 2
−p · q · θ ·
2·t
1
−
· q · (t · xA + t · (xA − 2)) (4 · s − 4 · p + 4 · z + t · xA + t · (xA − 2))
8·t
1
· s2 · (F + V · q · (t · xA − t · (xA − 2)))
−
2·t
Social surpluses for town A and B given one airport.
SSA1
SSB1
1
= · q · t − s2F ·
2
F
+ V · q − q · (p − z + t · xF ) + p · q · θ
2
1
F
2
= · q · (xF − 1) (2 · sF − 2 · p + t + 2 · z − t · xF ) − sF ·
+V ·q
2
2
1
− · q · (xF − 2) (2 · sF − 2 · p − 2 · t + 2 · z + t · xF ) + p · q · θ
2
28
Social surplus for the federal government given one airport
SSF1
1
F
2
= · q · t − 2 · sF ·
+ V · q − q · (p − sF − z + t · xF )
2
2
1
+ · q · (xF − 1) (2 · sF − 2 · p + t + 2 · z − t · xF )
2
1
+ · q · (xF − 2) (2 · xF − 2 · p − 2 · t + 2 · z + t · xF ) + 2 · p · q · θ
2
Conditions for maximum
The sucient conditions for a maximum is as follows
For two variables (Chiang and Wainwright, 2005, 298)
fxA xA < 0,
fxB xB < 0,
fxA xA fxB xB > fx2A xB
For one variable (Chiang and Wainwright, 2005, 292)
f 00 (x) < 0
Second derivatives of
SSF2
f xA xA =
∂ 2 SSF2
3·q·t
=−
<0
2
∂xA
2
f xB xB =
∂ 2 SSF2
3·q·t
=−
<0
2
∂xB
2
By Young's Theorem
29
(13)
f xA xB =
Second derivative of
3·q·t
−
2
∂ 2 SSF2
q·t
∂ 2 SSF2
=
=
∂xA xB
∂xB xA
2
2
3·q·t
q·t
−
>
2
2
SSF1
∂SSF1
= −2 · q · t < 0
xF
Second derivatives of
SSF2
and
SSF1
w.r.t.
sA , sB
and
sF
∂ 2 SSF2
∂ 2 SSF2
=
= −4 (F + V q) < 0
∂s2A
∂s2B
∂ 2 SSF1
= − (2F + 4V q) < 0
∂s2F
30
respectively